Using a 6wt rod and casting a beadhead wooly bugger seems to transform my casts into jerky, idiot-like throws. I don't expect to have the graceful loops that I would have if I were casting a small dry fly, but my casts when using heavier flies are downright ugly.

I'm assuming that the WB's are in size 2,4,or 6, (maybe even 8's but they would not have much excess weight). So with all the prior good advice, if you want to stay with the same line, floater, shorten and stiffen your leader, combined with larger loops in the cast. Shorten is easy, buy a 0X or 2X 7 foot leader and chop off 12 to 18 inches then add back 6 to 8 inches of 2X fluoro. If you want to go deeper, buy a Rio poly leader (7 foot long) with a sink rate of 4 IPS or 7 IPS depending on how deep you want to go. I make a small loop at the end of the poly leader and add 8 to 12 inches or 2X or 3X fluoro in a loop to loop connection.

I think even more important than than open loops, slowing your stroke, etc... Is controlling the timing & direction of your cast.

If you need to change direction from your pickup, strip in line to do it. Do not try and change direction with a load of line out. You will never establish control of the weighted fly.

When you establish your direction (backcast 180 degrees from target), start shooting line into your cast, but pay special attention to the weight at the end of your flyline. Let your cast extend COMPLETELY. You'll feel the fly hit the end of the line. You have to time your back and forward casts to the exact moment the fly hits the end of the line. That way, when you make your stroke (standard smooth casting stroke) your fly (weight) is already in line with your target and can be smoothly accelerated in that direction. If you start your stroke too early, you are pulling the fly around the corner and it's inertia is not in-line with the rest of your flyline. If you start too late (or over power your cast) the fly will bounce at the end of the line and lose it's ideal starting position. Think of someone standing behind you holding your fly while you load the rod. The fly and line are perfectly in-line with each other, and thus accelerate to exactly the same point. You'll know exactly what I'm talking about when you feel it. You'll also hit yourself in the back of the head a lot less.